Nonlinear dynamic modeling method and system for gear transmission system considering thermal deformation

Abstract

The invention discloses a gear transmission system nonlinear dynamic modeling method and system considering thermal deformation, and relates to the technical field of gear transmission system dynamic simulation. The method comprises the steps that component parameters and operation parameters of the planetary gear system are received; based on the component parameters and the operation parameters of the planetary gear system, a tooth surface flash temperature model is constructed, and a tooth surface friction coefficient is obtained through calculation; and constructing a tooth profile thermal deformation model, and respectively calculating internal excitation of the planetary gear system considering tooth profile thermal deformation. According to the method, the gear surface flash temperature model, the tooth profile thermal deformation model and the bearing gap model are constructed, the helical tooth planetary gear train-bearing nonlinear dynamic model is established, and the technical defect that the gear temperature rise effect and the nonlinear factors of the bearing cannot be fully considered in traditional helical tooth planetary gear train dynamic modeling is overcome; and the vibration response prediction precision under the high-speed and heavy-load working condition is obviously improved.

Description

Technical Field

[0001] The present invention relates to the technical field of gear transmission system dynamics simulation, and in particular to a nonlinear dynamics modeling method and system for a gear transmission system taking thermal deformation into consideration. Background Art

[0002] As a core power transmission device with high load capacity and high transmission efficiency, helical planetary gear trains are crucial in electric vehicle in-wheel drive systems. By integrating the in-wheel motor with a planetary reducer to directly drive the wheels, they offer advantages such as compact structure, efficient transmission, and rapid response, making them a key solution for lightweight and integrated electric vehicles. However, compared to traditional gearboxes, the heat sources in in-wheel drive systems are more complex. The combined effects of motor electromagnetic heat and gear friction heat result in localized system temperature rises far exceeding those of traditional gearboxes, further exacerbating thermal deformation of tooth surfaces and meshing misalignment. In this context, nonlinear issues caused by temperature rise are particularly prominent. Therefore, developing a refined dynamic model that comprehensively considers thermal deformation, tooth friction, and bearing clearance to address the unique needs of electric vehicle in-wheel drive systems is of great engineering value for revealing the mechanisms of thermally induced nonlinear vibrations and improving transmission system reliability.

[0003] Current research on the dynamic modeling of planetary gear systems often treats the temperature field as a constant parameter, or adopts a simplified processing method of equivalent thermal loads, which fails to effectively characterize the transient changes in tooth surface flash temperature and the dynamic evolution of thermal deformation. In addition, the existing dynamic modeling of planetary gear systems fails to consider the impact of the nonlinear characteristics of bearings on the system. Specifically, the existing tooth surface friction coefficient calculation model does not consider the changes in lubricating oil film characteristics caused by temperature rise, the calculation of tooth profile thermal deformation mostly relies on empirical formulas and lacks dynamic correlation with operating conditions, and the coupling mechanism of key excitation parameters such as time-varying meshing stiffness, tooth side clearance, meshing error, and bearing clearance caused by thermal deformation is insufficient. The nonlinear characteristics of bearing clearance are not fully considered in the modeling process. This leads to the limitations of traditional models in predicting the nonlinear vibration characteristics of gear systems under high-speed and heavy-load conditions, such as uncompensated thermal parameter drift and inaccurate multi-physics field coupling effects, making it difficult to meet the needs of high-precision dynamic simulation.

[0004] To this end, the present invention proposes a nonlinear dynamic modeling method and system for a gear transmission system taking thermal deformation into consideration. Summary of the Invention

[0005] The purpose of the present invention is to provide a nonlinear dynamic modeling method and system for a gear transmission system taking thermal deformation into consideration. This not only significantly improves the vibration response prediction accuracy under high-speed and heavy-load conditions, but also provides a refined simulation tool for the thermal-vibration coupling suppression design of helical planetary gear trains. At the same time, by establishing a mapping relationship between the temperature field and the dynamic reliability index, it provides a theoretical basis for the life prediction of the gear system and the evaluation of the adaptability to working conditions.

[0006] According to a first aspect of the present invention, in order to achieve the above-mentioned purpose, the present invention provides the following technical solution: a nonlinear dynamic modeling method of a gear transmission system considering thermal deformation, comprising:

[0007] receiving component parameters and operating parameters of the planetary gear system;

[0008] Based on the component parameters and operating parameters of the planetary gear system, a tooth surface flash temperature model is constructed and the tooth surface friction coefficient is calculated. The tooth surface friction force and friction torque are calculated based on the tooth surface friction coefficient.

[0009] Construct a tooth profile thermal deformation model and calculate the internal excitation of the planetary gear system considering the thermal deformation of the tooth profile, including meshing error calculation, time-varying meshing stiffness calculation, and tooth side clearance calculation;

[0010] Construct a bearing dynamics model and calculate the bearing clearance displacement function and bearing support force considering temperature rise;

[0011] The internal excitation considering the temperature rise effect is introduced into the planetary gear system, and combined with the bearing dynamics model, a nonlinear dynamic model of the helical planetary gear train-bearing system is established.

[0012] Furthermore, the planetary gear system includes a sun gear, planet gears, an inner ring gear, a planet carrier and bearings;

[0013] The basic parameters of the planetary gear system include mass, number of planetary gears, number of teeth on the inner ring gear, module of the inner ring gear, tooth width, pitch circle radius, helix angle, pressure angle, base circle radius of each gear, and moment of inertia of each gear;

[0014] The operating conditions of the planetary gear system include input speed and torque, initial temperature, and operating temperature.

[0015] Furthermore, based on the component parameters and operating parameters of the planetary gear system, a tooth surface flash temperature model is constructed to calculate the tooth surface friction coefficient. The tooth surface friction coefficient includes the tooth surface friction force and friction torque, as follows:

[0016] The calculation model of tooth surface instantaneous flash temperature is obtained based on Blok flash temperature theory:

[0017]

[0018] Where u is the friction coefficient, f m is the temperature rise coefficient, f e is the normal load per unit tooth width on the tooth surface, v1 and v2 are the tangential velocities of the meshing line of the driving wheel and the driven wheel respectively, g1 and g2 are the heat transfer coefficients of the driving wheel and the driven wheel respectively, ρ1 and ρ2 are the material densities of the two gears respectively, c1 and c2 are the specific heat capacities of the two gear materials respectively, and B is half of the contact band width;

[0019] The tangential velocity calculation expression of the meshing line between the driving wheel and the driven wheel is as follows:

[0020]

[0021] Where, ω i Represents the angular velocity of the driving wheel and the driven wheel, r i They represent the pitch circle radius of the driving wheel and the driven wheel respectively, α represents the initial pressure angle of the gear, r ki is the distance from the meshing point to the center of the driving wheel and the driven wheel, where 1 represents the driving wheel and 2 represents the driven wheel;

[0022] The tooth surface friction coefficient refers to the ratio of friction force to normal pressure during gear meshing. Due to differences in lubrication conditions, the friction coefficient will change dynamically. In actual high-temperature and high-speed working conditions, the planetary gear system is in a mixed lubrication state. Mixed lubrication states include dry friction, boundary lubrication, mixed lubrication, and elastohydrodynamic lubrication. The tooth surface friction coefficient in the mixed lubrication state is:

[0023]

[0024] in,

[0025]

[0026] v s (t)=|v1(t)-v2(t)|

[0027] Where S av is the average value of tooth surface roughness, S1 and S2 are the surface roughness of the driving wheel and the driven wheel respectively, P ei (t) is the normal load on the gear tooth width, T is the total load on the tooth surface, b i are the tooth widths of the driving wheel and the driven wheel, r bi are the base circle radii of the driving wheel and the driven wheel respectively, are the entrainment speeds of the driving wheel and the driven wheel, v s (t) is the relative sliding speed between the driving wheel and the driven wheel, and η0 is the dynamic viscosity coefficient of the lubricating oil.

[0028] Furthermore, a tooth profile thermal deformation model is constructed to calculate the internal excitation of the planetary gear system considering the thermal deformation of the tooth profile, including the meshing error calculation, time-varying meshing stiffness calculation, and tooth side clearance calculation. Specifically,

[0029] (41) The tooth profile thermal deformation calculation model is a gear thermal deformation prediction method established based on the principles of thermodynamics and material mechanics, combined with numerical simulation technology. By analyzing the thermal expansion characteristics and temperature field distribution law of gear materials at high temperatures, it reflects the nonlinear relationship between the material linear expansion coefficient and the temperature gradient. The calculation formula for tooth profile thermal deformation is as follows:

[0030] Δf=Δlcosα=Δθ k r k cosα

[0031] in,

[0032]

[0033] Where Δf is the thermal deformation of the tooth profile, Δl is the radius r k , the angle is Δθ k An arc, α is the initial pressure angle, Δθ k is the angle between the meshing point and the gear center before and after deformation, r k is the distance between the meshing point and the gear center before deformation, λ is the thermal expansion coefficient, T(r k ) is the instantaneous contact temperature at the meshing point, T0 is the initial temperature, s is the tooth thickness at the pitch circle, r is the pitch circle radius, α k is the pressure angle of the meshing point after deformation, r k' is the distance between the meshing point and the gear center after deformation, r b is the base circle radius, u b is the distance between the pitch circle and the base circle, r x is the radial distance, T(r k ) is the radial distance r x The instantaneous contact temperature, r a is the radial distance of the tooth tip, r s is the radial distance at the pitch circle;

[0034] (42) Meshing error excitation calculation considering thermal deformation: When the planetary gear system operates under high temperature conditions, the significant increase in tooth surface temperature will trigger the thermal deformation effect of the material, which in turn leads to the nonlinear offset of the involute geometric characteristics of the tooth profile. The tooth profile error caused by the temperature increase of the tooth surface is expressed as:

[0035]

[0036] Where s gi (i=1,2) represents the tooth thickness of the two gears. The external meshing pair driving gear is the sun gear, and the internal meshing pair driving gear is the planet gear. bi Indicates the base circle radius of the two gears, r ci represents the curvature radius of the two gears, u bi is the thermal expansion deformation of the two gears, αki is the pressure angle of the contact point after the gear is thermally expanded;

[0037] The comprehensive tooth profile deviation considering the tooth surface temperature rise and manufacturing error can be expressed as:

[0038]

[0039] Where, represents the initial tooth profile error of the two gears due to manufacturing error, (G T+ΔT ) 1,2 Indicates the deviation of the tooth profiles of the two gears caused by temperature rise;

[0040] The obtained comprehensive tooth profile deviation is equivalent to the comprehensive equivalent error along the meshing line of the gear pair:

[0041]

[0042] The meshing error excitation of the planetary gear system is obtained as:

[0043]

[0044] Where, E s 、E pi are the eccentricity errors of the sun gear and the i-th planet gear, ω s 、ω pi 、ω c are the rotation frequencies of the sun gear, the i-th planet gear and the planet carrier, β s , β pi are the initial phases of the eccentricity errors between the sun gear and the i-th planet gear, φ pi is the position angle of the i-th planetary gear, β b is the helix angle;

[0045] (43) Calculation of gear meshing deformation coordination conditions considering thermal deformation: Based on the elastic deformation characteristics of the internal and external meshing in the planetary gear system, a deformation coordination equation considering material elastic deformation, meshing error and vibration displacement is established to solve the relative displacement of the internal and external gear meshing pairs on the meshing line. The equivalent displacement of the sun gear and planetary gear meshing pairs is expressed by δ spi express:

[0046] δ spi

[0047] =(-x s sinψ spi +y s cosψ spi +u s +x pi sinψ spi -y pi cosψspi +u pi )

[0048] cosβ b +(z s -z pi )sinβ b -e spi

[0049] Where, ψ spi is the angle between the meshing plane of the external meshing pair and the y-axis, β b is the helix angle, e spi is the meshing error of the external gear pair;

[0050] The equivalent displacement of the meshing pair of planetary gear and internal gear ring is expressed as δ rpi express:

[0051] δ rpi

[0052] =(x r sinψ rpi -y r cosψ rpi +u r -x pi sinψ rpi +y pi cosψ rpi -u pi )cosβ b +(z pi -z r )sinβ b -e rpi

[0053] Where x s 、y s 、z s and u s are the displacement of the sun gear along the coordinate system and the torsion angle around the z axis, x pi 、y pi 、z pi and u pi are the displacement of the planetary gear along the coordinate system and the torsion angle around the z axis, ψ rpi is the angle between the meshing plane of the internal meshing pair and the y-axis, e rpi Meshing error of internal gear pair;

[0054] The equivalent displacement projection of the planet gear and the planet carrier along the planet carrier coordinate system is:

[0055]

[0056] Where, δ cpix , δ cpiy , δcpiz and δ cpiu are the projections of the equivalent displacement of the planetary gear along the planetary carrier coordinate system, x c 、y c 、z c and u c are the displacement of the planet carrier along the coordinate system and the torsion angle around the z axis, φ pi is the planet gear position angle;

[0057] (44) Calculation of time-varying mesh stiffness considering gear thermal deformation: The potential energy method is used to calculate the time-varying mesh stiffness of gears;

[0058] (44.1) During the meshing process, a pair of gears will experience single-tooth meshing and double-tooth meshing respectively. When the gear is in the single-tooth meshing region, the calculation formula for the time-varying mesh stiffness is:

[0059]

[0060] Where k eh (t) is the Hertzian contact stiffness, k ebi (t)(i=1,2) represents the bending stiffness of the driving wheel and the driven wheel respectively, k esi (t)(i=1,2) represents the shear stiffness of the driving wheel and the driven wheel respectively, k eai (t)(i=1,2) represents the radial compression stiffness of the driving wheel and the driven wheel respectively, k efi (t) (i=1, 2) represents the matrix stiffness of the driving wheel and the driven wheel respectively;

[0061] (44.2) When the gear is in the double-tooth meshing region, the calculation formula for the time-varying mesh stiffness is:

[0062]

[0063] Where k eh,j (t)(j=1, 2) respectively represent the Hertzian contact stiffness of the two meshing pairs when the driving wheel and the driven wheel are in the double-tooth meshing area;

[0064] As the gear temperature rises during operation, the instantaneous contact temperature of the tooth surface changes, causing the tooth surface profile to deform and the meshing stiffness to change. According to Hertz contact theory, the stiffness change of the driving and driven wheels caused by the instantaneous contact temperature change of the gear surface is:

[0065]

[0066] Where k Ti (i=1, 2) represents the change of the tooth surface stiffness of the driving wheel and the driven wheel respectively, b i(i=1, 2) represents the tooth width of the driving wheel and the driven wheel respectively, Δf is the thermal deformation of the tooth profile, F n is the tooth surface pressure, and the stiffness changes of the two tooth surfaces in series can be obtained by the equivalent stiffness k caused by temperature change. T (t);

[0067] Assuming that the thermal deformation of the internal and external meshing of the planetary gear train is consistent, the calculation formula for the time-varying meshing stiffness considering the thermal deformation of the gear can be obtained as follows:

[0068] k spi (t) = k rpi (t) = k e (t)+k T (t)

[0069] Where k spi (t) is the time-varying meshing stiffness of the planetary gear system, k rpi (t) is the time-varying meshing stiffness of the planetary gear train, k e (t) is the time-varying mesh stiffness without considering the temperature rise, k T (t) is the equivalent stiffness caused by temperature change;

[0070] (45) Calculation of tooth side clearance considering thermal deformation: tooth side clearance refers to the minimum clearance between the two gear tooth sides in the direction perpendicular to the axis when the gear pair is in the non-meshing state. Since the gear teeth will produce thermal deformation during the transmission process, the tooth side clearance of the gear will decrease. Therefore, after considering the influence of thermal deformation, the tooth side clearance value calculation formula is:

[0071]

[0072] Where b' is half of the tooth backlash value after considering thermal deformation, b is half of the initial value of the tooth backlash, Δf1 and Δf2 are the thermal deformation of the tooth profile of the driving wheel and the driven wheel respectively;

[0073] The calculation formula for tooth side clearance taking thermal deformation into account is:

[0074]

[0075] Where x is the displacement along the gear meshing line.

[0076] Furthermore, a bearing dynamics model was constructed to calculate the bearing clearance displacement function and bearing support force considering the temperature rise, as follows:

[0077] The nonlinear displacement function of the bearing clearance considering the thermal deformation of the rolling elements is:

[0078]

[0079] in,

[0080] b bi =b b0 -Δf b

[0081] Where x bi with y bi The clearance displacement of the two bearings along the two directions of the coordinate system, b bi is half of the clearance value after thermal deformation of the two bearings, b b0 is half of the initial bearing clearance value, Δf b is the thermal deformation of the rolling element;

[0082] The bearing support force is the resultant of the forces acting on each roller. The resultant bearing support forces in the x and y directions are obtained by projecting the normal force at the contact point of each rolling element into its radial direction and then solving and summing the radial forces of all rolling elements in the x and y directions.

[0083] The forces acting on each rolling element are calculated using Hertz contact theory:

[0084]

[0085] Where K b is the rolling element stiffness, Δ bk is the deformation of the kth rolling element, H(Δ bk ) is the Hertzian contact function;

[0086] Decompose each rolling element load along the x and y directions, and sum them to obtain the bearing support force along the two directions:

[0087]

[0088] Where, ψ' b is the contact angle, θ bk is the angular displacement of the kth rolling element.

[0089] Furthermore, in the nonlinear dynamic model of the helical planetary gear train-bearing system, both the internal gear and the external gear are involute helical gears, each component is simplified as a rigid body, and the installation errors between the components are ignored; the contact mode of the joint surface is simplified to a spring and damping connection;

[0090] The planetary gears are evenly distributed along the circumference, and the average meshing stiffness, support stiffness, mass, moment of inertia and structural dimensions of the planetary gears are all equal.

[0091] According to a second aspect of the present invention, the present invention provides a nonlinear dynamic modeling system for a gear transmission system considering thermal deformation, which is used to implement the above-mentioned nonlinear dynamic modeling method for a gear transmission system considering thermal deformation, comprising:

[0092] A receiving module, configured to receive component parameters and operating parameters of the planetary gear system;

[0093] The tooth surface flash temperature model construction module is used to construct the tooth surface flash temperature model and calculate the tooth surface friction coefficient based on the component parameters and operating parameters of the planetary gear system, and calculate the tooth surface friction force and friction torque based on the tooth surface friction coefficient;

[0094] The tooth profile thermal deformation model construction module is used to construct the tooth profile thermal deformation model and calculate the internal excitation of the planetary gear system considering the thermal deformation of the tooth profile, including the meshing error calculation, time-varying meshing stiffness calculation, and tooth side clearance calculation;

[0095] The bearing dynamics model construction module is used to construct the bearing dynamics model and calculate the bearing clearance displacement function and bearing support force considering the temperature rise;

[0096] The nonlinear dynamic model building module is used to introduce internal excitation considering the temperature rise effect into the planetary gear system, and combine it with the bearing dynamic model to establish a nonlinear dynamic model of the helical planetary gear train-bearing system.

[0097] According to the third aspect of the present invention, the present invention provides a terminal device, including a memory, a processor, and a computer program stored in the memory and capable of running on the processor. When the processor loads and executes the computer program, the above-mentioned nonlinear dynamic modeling method of the gear transmission system considering thermal deformation is adopted.

[0098] According to a fourth aspect of the present invention, the present invention provides a storage medium containing computer executable instructions, which, when executed by a computer processor, are used to perform the above-mentioned nonlinear dynamic modeling method of a gear transmission system considering thermal deformation.

[0099] According to a fifth aspect of the present invention, the present invention provides a computer program product, which includes a computer program. When the computer program is executed by a processor, it is used to load and execute the above-mentioned nonlinear dynamic modeling method of the gear transmission system considering thermal deformation.

[0100] The present invention has at least the following beneficial effects:

[0101] 1. This invention establishes a nonlinear dynamic model of a helical planetary gear train and bearing by constructing models of tooth surface flash temperature, tooth profile thermal deformation, and bearing clearance. This systematically addresses the technical deficiency of traditional helical planetary gear train dynamic modeling, which fails to fully consider the temperature rise effect of gears and the nonlinear factors of bearings. By accurately analyzing the dynamic evolution of the tooth surface friction coefficient with temperature rise and the quantitative correlation between the thermal deformation of the tooth profile and the operating conditions, this invention achieves thermal coupling correction of key internal excitation parameters such as meshing error, time-varying meshing stiffness, tooth side clearance, and bearing clearance, and establishes a nonlinear dynamic equation that includes the thermoelastic coupling effect.

[0102] 2. Based on the traditional nonlinear dynamic model of the planetary gear system, the present invention innovatively introduces a bearing system, fully considers the influence of the temperature rise effect on the bearing clearance, solves the radial bearing support force, and establishes a nonlinear dynamic model of the bearing. The calculation of the bearing displacement needs to consider the displacement of the sun gear. Conversely, the existence of the bearing support force also affects the support stiffness of the sun gear.

[0103] 3. This invention not only significantly improves the vibration response prediction accuracy under high-speed and heavy-load conditions, but also provides a refined simulation tool for the thermal-vibration coupling suppression design of helical planetary gear trains. At the same time, by establishing a mapping relationship between temperature field and dynamic reliability indicators, it provides a theoretical basis for gear system life prediction and working condition adaptability evaluation.

[0104] Of course, any product implementing the present invention does not necessarily need to achieve all of the advantages described above at the same time. BRIEF DESCRIPTION OF THE DRAWINGS

[0105] Figure 1 Schematic diagram of the process of the modeling method of the present invention;

[0106] Figure 2 This is a schematic diagram of the framework of the modeling method of the present invention;

[0107] Figure 3 Schematic diagram of the tooth profile thermal deformation model of the present invention;

[0108] Figure 4 Schematic diagram of the helical planetary gear train-bearing nonlinear dynamic model of the present invention. DETAILED DESCRIPTION

[0109] The following will be combined with the accompanying drawings in the embodiments of the present disclosure to clearly and completely describe the technical solutions in the embodiments of the present disclosure. Obviously, the embodiments described are only part of the embodiments of the present disclosure, not all of the embodiments. Based on the embodiments of the present disclosure, all other embodiments obtained by ordinary technicians in this field without making any creative efforts are within the scope of protection of the present disclosure.

[0110] Example 1:

[0111] See also Figures 1-4 The present invention provides a technical solution: a nonlinear dynamic modeling method for a gear transmission system considering thermal deformation, comprising the following steps:

[0112] S1. Receive component parameters and operating parameters of the planetary gear system;

[0113] The planetary gear system includes a sun gear, planet gears, an internal gear ring, a planet carrier and bearings;

[0114] The basic parameters of the planetary gear system include mass, number of planetary gears, number of teeth on the inner ring gear, module of the inner ring gear, tooth width, pitch circle radius, helix angle, pressure angle, base circle radius of each gear, and moment of inertia of each gear;

[0115] The operating conditions of the planetary gear system include input speed and torque, initial temperature, and operating temperature;

[0116] S2. Based on the component parameters and operating parameters of the planetary gear system, construct a tooth surface flash temperature model and calculate the tooth surface friction coefficient, which includes the tooth surface friction force and friction torque;

[0117] The meshing modes in the planetary gear system include internal meshing and external meshing. For internal meshing, the driving gear is the sun gear and the driven gear is the planet gear; for external meshing, the driving gear is the planet gear and the driven gear is the inner ring gear.

[0118] The calculation model of tooth surface instantaneous flash temperature is obtained based on Blok flash temperature theory:

[0119]

[0120] Where u is the friction coefficient, f m is the temperature rise coefficient, f e is the normal load per unit tooth width on the tooth surface, v1 and v2 are the tangential velocities of the meshing line of the driving wheel and the driven wheel respectively, g1 and g2 are the heat transfer coefficients of the driving wheel and the driven wheel respectively, ρ1 and ρ2 are the material densities of the two gears respectively, c1 and c2 are the specific heat capacities of the two gear materials respectively, and B is half of the contact band width;

[0121] The tangential velocity calculation expression of the meshing line between the driving wheel and the driven wheel is as follows:

[0122]

[0123] Where, ω i Represents the angular velocity of the driving wheel and the driven wheel, r i They represent the pitch circle radius of the driving wheel and the driven wheel respectively, α represents the initial pressure angle of the gear, r ki is the distance from the meshing point to the center of the driving wheel and the driven wheel, where 1 represents the driving wheel and 2 represents the driven wheel;

[0124] The tooth surface friction coefficient refers to the ratio of friction force to normal pressure during gear meshing. Its value directly affects transmission efficiency, wear, and noise. Due to differences in lubrication conditions, the friction coefficient will change dynamically. Under actual high-temperature and high-speed working conditions, the planetary gear system is in a mixed lubrication state. Mixed lubrication states include dry friction, boundary lubrication, mixed lubrication, and elastohydrodynamic lubrication. The tooth surface friction coefficient under mixed lubrication conditions is:

[0125]

[0126] in,

[0127]

[0128] v s (t)=|v1(t)-v2(t)|

[0129] Where S av is the average value of tooth surface roughness, S1 and S2 are the surface roughness of the driving wheel and the driven wheel respectively, P ei (t) is the normal load on the gear tooth width, T is the total load on the tooth surface, b i are the tooth widths of the driving wheel and the driven wheel, r bi are the base circle radii of the driving wheel and the driven wheel respectively, are the entrainment speeds of the driving wheel and the driven wheel, v s (t) is the relative sliding speed between the driving wheel and the driven wheel, η0 is the dynamic viscosity coefficient of the lubricating oil;

[0130] S3. Construct a tooth profile thermal deformation model and calculate the internal excitation of the planetary gear system considering the thermal deformation of the tooth profile, including the meshing error calculation, time-varying meshing stiffness calculation, and tooth side clearance calculation, as follows:

[0131] (S31) The tooth profile thermal deformation calculation model is a gear thermal deformation prediction method established based on the principles of thermodynamics and material mechanics, combined with numerical simulation technology. By analyzing the thermal expansion characteristics and temperature field distribution law of gear materials at high temperatures, it reflects the nonlinear relationship between the material linear expansion coefficient and the temperature gradient. The calculation formula for tooth profile thermal deformation is as follows:

[0132] Δf=Δlcosα=Δθ k r k cosα

[0133] in,

[0134]

[0135] Where Δf is the thermal deformation of the tooth profile, Δl is the radius r k , the angle is Δθk An arc (see attached Figure 2 ), α is the initial pressure angle, Δθ k is the angle between the meshing point and the gear center before and after deformation (see Appendix Figure 2 ), r k is the distance between the meshing point and the gear center before deformation (see Appendix Figure 2 ), thermal expansion coefficient of λ, T(r k ) is the instantaneous contact temperature at the meshing point, T0 is the initial temperature, s is the tooth thickness at the pitch circle, r is the pitch circle radius, α k is the pressure angle of the meshing point after deformation, r k' is the distance between the meshing point and the gear center after deformation (see Appendix Figure 2 ), r b is the base circle radius, u b is the distance between the pitch circle and the base circle, r x is the radial distance, T(r k ) is the radial distance r x The instantaneous contact temperature, r a is the radial distance of the tooth tip, r s is the radial distance at the pitch circle;

[0136] (S32) Meshing error excitation calculation considering thermal deformation: When a planetary gear system operates under high temperature conditions, a significant increase in tooth surface temperature will induce thermal deformation effects on the material, which in turn leads to nonlinear deviations in the geometric characteristics of the tooth profile involute. The tooth profile error caused by the temperature increase of the tooth surface is expressed as:

[0137]

[0138] Where s gi (i=1,2) represents the tooth thickness of the two gears. The external meshing pair driving gear is the sun gear, and the internal meshing pair driving gear is the planet gear. bi Indicates the base circle radius of the two gears, r ci represents the curvature radius of the two gears, u bi is the thermal expansion deformation of the two gears, α ki is the pressure angle of the contact point after the gear is thermally expanded;

[0139] The comprehensive tooth profile deviation considering the tooth surface temperature rise and manufacturing error can be expressed as:

[0140]

[0141] Where, represents the initial tooth profile error of the two gears due to manufacturing error, (G T+ΔT ) 1,2 Indicates the deviation of the tooth profiles of the two gears caused by temperature rise;

[0142] The obtained comprehensive tooth profile deviation is equivalent to the comprehensive equivalent error along the meshing line of the gear pair:

[0143]

[0144] The meshing error excitation of the planetary gear system is obtained as:

[0145]

[0146] Where, E s 、E pi are the eccentricity errors of the sun gear and the i-th planet gear, ω s 、ω pi 、ω c

[0147] are the rotation frequencies of the sun gear, the i-th planet gear and the planet carrier, β s , β pi are the initial phases of the eccentricity errors between the sun gear and the i-th planet gear, φ pi is the position angle of the i-th planetary gear, β b is the helix angle;

[0148] Calculation of gear meshing deformation coordination conditions considering thermal deformation: Based on the elastic deformation characteristics of the internal and external meshing in the planetary gear system, a deformation coordination equation is established that takes into account the elastic deformation of the material, meshing error, and vibration displacement. The relative displacement of the internal and external gear meshing pairs on the meshing line is solved, and the equivalent displacement of the sun gear and planetary gear meshing pairs is expressed by δ spi express:

[0149] δ spi

[0150] =(-x s sinψ spi +y s cosψ spi +u s +x pi sinψ spi -y pi cosψ spi +u pi )

[0151] cosβ b +(z s -z pi )sinβ b -e spi

[0152] Where x s 、y s 、z s and u sare the displacement of the sun gear along the coordinate system and the torsion angle around the z axis, x pi 、y pi 、z pi and u pi are the displacement of the planetary gear along the coordinate system and the torsion angle around the z axis, ψ spi is the angle between the meshing plane of the external meshing pair and the y-axis, β b is the helix angle, e spi is the meshing error of the external gear pair;

[0153] The equivalent displacement of the meshing pair of planetary gear and internal gear ring is expressed as δ rpi express:

[0154] δ rpi

[0155] =(x r sinψ rpi -y r cosψ rpi +u r -x pi sinψ rpi +y pi cosψ rpi -u pi )cosβ b +(z pi -z r )sinβ b -e rpi

[0156] Where, ψ rpi is the angle between the meshing plane of the internal meshing pair and the y-axis, e rpi Meshing error of internal gear pair;

[0157] The equivalent displacement projection of the planet gear and the planet carrier along the planet carrier coordinate system is:

[0158]

[0159] Where, δ cpix , δ cpiy , δ cpiz and δ cpiu are the projections of the equivalent displacement of the planetary gear along the planetary carrier coordinate system, x c 、y c 、z c and u c are the displacement of the planet carrier along the coordinate system and the torsion angle around the z axis, φ pi is the planet gear position angle;

[0160] (S33) Calculation of time-varying mesh stiffness considering thermal deformation of gears: Common methods for calculating time-varying mesh stiffness of gears include: potential energy method, finite element numerical calculation method, and experimental method. The potential energy method calculates time-varying mesh stiffness based on the potential energy principle and related theories of material mechanics, and has fast calculation speed and high calculation accuracy. This embodiment uses the potential energy method to calculate the time-varying mesh stiffness of gears;

[0161] (S33.1) The principle of the potential energy method is briefly summarized, focusing on the effect of gear thermal deformation on time-varying mesh stiffness;

[0162] A pair of gears will experience single-tooth meshing and double-tooth meshing during the meshing process. When the gear is in the single-tooth meshing area, the calculation formula for the time-varying meshing stiffness is:

[0163]

[0164] Where k eh (t) is the Hertzian contact stiffness, k ebi (t)(i=1,2) represents the bending stiffness of the driving wheel and the driven wheel respectively, k esi (t)(i=1,2) represents the shear stiffness of the driving wheel and the driven wheel respectively, k eai (t)(i=1,2) represents the radial compression stiffness of the driving wheel and the driven wheel respectively, k efi (t) (i=1, 2) represents the matrix stiffness of the driving wheel and the driven wheel respectively;

[0165] (S33.2) When the gear is in the double-tooth meshing region, the calculation formula for the time-varying mesh stiffness is:

[0166]

[0167] Where k eh,j (t)(j=1, 2) respectively represent the Hertzian contact stiffness of the two meshing pairs when the driving wheel and the driven wheel are in the double-tooth meshing area;

[0168] As the gear temperature rises during operation, the instantaneous contact temperature of the tooth surface changes, causing the tooth surface profile to deform and the meshing stiffness to change. According to Hertz contact theory, the stiffness change of the driving and driven wheels caused by the instantaneous contact temperature change of the gear surface is:

[0169]

[0170] Where k Ti (i=1, 2) represents the change of the tooth surface stiffness of the driving wheel and the driven wheel respectively, b i (i=1, 2) represents the tooth width of the driving wheel and the driven wheel respectively, Δf is the thermal deformation of the tooth profile, F nis the tooth surface pressure, and the stiffness changes of the two tooth surfaces in series can be obtained by the equivalent stiffness k caused by temperature change. T (t);

[0171] Assuming that the thermal deformation of the internal and external meshing of the planetary gear train is consistent, the calculation formula for the time-varying meshing stiffness considering the thermal deformation of the gear can be obtained as follows:

[0172] k spi (t) = k rpi (t) = k e (t)+k T (t)

[0173] Where k spi (t) is the time-varying meshing stiffness of the planetary gear system, k rpi (t) is the time-varying meshing stiffness of the planetary gear train, k e (t) is the time-varying mesh stiffness without considering the temperature rise, k T (t) is the equivalent stiffness caused by temperature change;

[0174] (S34) Calculation of tooth side clearance considering thermal deformation: Tooth side clearance refers to the minimum clearance between the tooth flanks of two gears in the non-meshing state, perpendicular to the axis. It is primarily used to compensate for manufacturing and installation errors, cushion vibration, and accommodate thermal expansion. Its size directly affects the performance of the transmission system: Too small a clearance will lead to overtightening of the tooth surfaces, exacerbating friction, wear, and meshing shock, and causing high-frequency vibration; too large a clearance will reduce the contact ratio, increase transmission errors and noise, and affect accuracy.

[0175] Since the gear teeth will produce thermal deformation during the transmission process, the gear tooth side clearance will decrease. Therefore, after considering the influence of thermal deformation, the tooth side clearance value calculation formula is:

[0176]

[0177] Where b' is half of the tooth backlash value after considering thermal deformation, b is half of the initial value of the tooth backlash, Δf1 and Δf2 are the thermal deformation of the tooth profile of the driving wheel and the driven wheel respectively;

[0178] The calculation formula for tooth side clearance taking thermal deformation into account is:

[0179]

[0180] Where x is the displacement along the gear meshing line;

[0181] S4. Construct a bearing dynamics model and calculate the bearing clearance displacement function and bearing support force considering the temperature rise, as follows:

[0182] The nonlinear displacement function of the bearing clearance considering the thermal deformation of the rolling elements is:

[0183]

[0184]

[0185] in,

[0186] b bi =b b0 -Δf b

[0187] Where x bi with y bi The clearance displacement of the two bearings along the two directions of the coordinate system, b bi is half of the clearance value after thermal deformation of the two bearings, b b0 is half of the initial bearing clearance value, Δf b is the thermal deformation of the rolling element;

[0188] The bearing support force is the resultant of the forces acting on each roller. The resultant bearing support forces in the x and y directions are obtained by projecting the normal force at the contact point of each rolling element into its radial direction and then solving and summing the radial forces of all rolling elements in the x and y directions.

[0189] The forces acting on each rolling element are calculated using Hertz contact theory:

[0190]

[0191] Where K b is the rolling element stiffness, Δ bk is the deformation of the kth rolling element, H(Δ bk ) is the Hertzian contact function;

[0192] Decompose each rolling element load along the x and y directions, and sum them to obtain the bearing support force along the two directions:

[0193]

[0194] Where, ψ' b is the contact angle, θ bk is the angular displacement of the kth rolling element;

[0195] S5. Introduce internal excitation into the planetary gear system taking into account the temperature rise effect and, combined with the bearing dynamics model, establish a nonlinear dynamic model of the helical planetary gear train-bearing system.

[0196] In the nonlinear dynamic model of the helical planetary gear train-bearing system, both the internal gear and the external gear are involute helical gears, each component is simplified as a rigid body, and the installation errors between the components are ignored; the contact mode of the joint surface is simplified to a spring and damper connection;

[0197] The planetary gears are evenly distributed along the circumference, and the average meshing stiffness, support stiffness, mass, moment of inertia and structural dimensions of the planetary gears are all equal.

[0198] Next, the present invention will be further described with reference to specific embodiments:

[0199] In this implementation case, the nonlinear dynamic differential equation of the established system is as follows:

[0200] The differential equation of motion of the sun gear is:

[0201]

[0202] Where m s is the mass of the sun gear, I s is the sun gear moment of inertia x s 、y s 、z s and u s are the displacement and torsion angle of the sun gear along the three directions of the coordinate system, k sx 、k sy 、k sz and k su are the support stiffness and torsional stiffness of the sun gear in different directions, c sx 、c sy 、c sz and c su are the support damping and torsional damping of the sun gear in different directions, F fs (t) is the friction force on the sun gear, T fs (t) is the friction torque on the sun gear, r bs is the sun gear base circle radius;

[0203] The differential equation of motion of the internal gear ring is:

[0204]

[0205] Where m r is the mass of the inner ring gear, I r is the moment of inertia of the inner gear ring, x r 、y r 、z r and u r They are the displacement and torsion angle of the inner gear ring along the three directions of the coordinate system, k rx 、k ry 、k rzand k ru are the support stiffness and torsional stiffness of the inner gear ring in different directions, c rx 、c ry 、c rz and c ru are the support damping and torsional damping of the inner gear ring in different directions, F fr (t) is the friction force on the inner gear ring, T fr (t) is the friction torque on the inner gear ring, r br is the base circle radius of the inner gear ring;

[0206] The differential equation of motion of the i-th planetary gear is:

[0207]

[0208] Where m pi is the mass of the i-th planetary gear, I pi is the moment of inertia of the i-th planetary gear, x pi 、y pi 、z pi and u pi They are the displacement of the planetary gear along the three directions of the coordinate system and the torsion angle around the z axis, k pix 、k piy 、k piz and k piu are the support stiffness and torsional stiffness of the planetary gear in different directions, c pix 、c piy 、c piz and c piu are the support damping and torsional damping of the planetary gear in different directions, r bpi is the planetary gear base circle radius;

[0209] The differential equation of motion of the planet carrier is:

[0210]

[0211] Where m c is the mass of the planet carrier, I c is the planet carrier moment of inertia, x c 、y c 、z c and u c They are the displacement of the planet carrier in three directions along the coordinate system and the torsion angle around the z axis, k cx 、k cy 、k cz and k cu are the support stiffness and torsional stiffness of the planet carrier in different directions, c cx 、c cy 、c czand c cu are the support damping and torsional damping of the planet carrier in different directions, r c is the planet carrier radius, T c Output torque to the planet carrier.

[0212] The differential equation of motion of the bearing is:

[0213]

[0214] Where m b1 With m b2 are the masses of the two bearings, x b1 with y b1 are the displacements of bearing 1 in two directions along the coordinate system, x b2 with y b2 are the displacements of bearing 2 in the two directions of the coordinate system, k b1x With k b2x are the stiffness of the two bearings along the x-axis, k b1y With k b2y are the stiffness of the two bearings along the y direction of the coordinate axis, c b1x with c b2x are the damping of the two bearings along the x-direction of the coordinate axis, c b1y with c b2y are the damping of the two bearings along the y direction of the coordinate axis, F b1x With F b2x are the supporting force components of the two bearings along the x-direction of the coordinate axis, F b1y With F b2y are the supporting force components of the two bearings along the y direction of the coordinate axis;

[0215] Assuming that the number of planetary gears is N, the differential equations of each subsystem in the planetary gear train (sun gear, planetary gear, ring gear, planet carrier and bearing) are combined to obtain the dynamic equations of the planetary gear train with 4N+16 degrees of freedom. The system degrees of freedom are expressed by the matrix X:

[0216] X=

[0217] (x s ,y s ,z s ,u s ,x c ,y c ,z c ,u c ,x r ,y r ,z r ,u r ,x b1 ,y b1 ,xb2 ,y b2 ,x p1 ,y p1 ,z p1 ,u p1 …,x pN ,y pN ,z pN ,u pN ) T .

[0218] In summary, the present invention introduces the temperature rise effect system into the nonlinear dynamic modeling system of the helical planetary gear system by constructing a calculation model of tooth surface flash temperature and tooth profile thermal deformation; by establishing a tooth surface flash temperature model to accurately characterize the dynamic evolution law of the friction coefficient with the temperature field, the thermoelastic theory is used to analyze the quantitative relationship between the thermal deformation of the tooth profile and the operating parameters, and the influence mechanism of temperature rise excitation and meshing error, time-varying meshing stiffness, tooth side clearance and bearing clearance is established; the modeling method of the present invention fully considers the influence of thermal deformation effect, combines the nonlinear dynamic model of the bearing, and establishes a helical planetary gear system-bearing nonlinear dynamic model, thereby improving the system modeling accuracy and providing theoretical support for the thermal-vibration coupling suppression design and dynamic reliability evaluation of the helical planetary gear system.

[0219] Example 2:

[0220] The present invention provides a nonlinear dynamic modeling system for a gear transmission system considering thermal deformation, which is used to implement the above-mentioned nonlinear dynamic modeling method for a gear transmission system considering thermal deformation, comprising:

[0221] A receiving module, configured to receive component parameters and operating parameters of the planetary gear system;

[0222] The tooth surface flash temperature model construction module is used to construct the tooth surface flash temperature model and calculate the tooth surface friction coefficient based on the component parameters and operating parameters of the planetary gear system. The tooth surface friction coefficient includes the tooth surface friction force and friction torque;

[0223] The tooth profile thermal deformation model construction module is used to construct the tooth profile thermal deformation model and calculate the internal excitation of the planetary gear system considering the thermal deformation of the tooth profile, including the meshing error calculation, time-varying meshing stiffness calculation, and tooth side clearance calculation;

[0224] The bearing dynamics model construction module is used to construct the bearing dynamics model and calculate the bearing clearance displacement function and bearing support force considering the temperature rise;

[0225] The nonlinear dynamic model building module is used to introduce internal excitation considering the temperature rise effect into the planetary gear system, and combine it with the bearing dynamic model to establish a nonlinear dynamic model of the helical planetary gear train-bearing system.

[0226] Specifically, the above-mentioned receiving module, tooth surface flash temperature model construction module, tooth profile thermal deformation model construction module, bearing dynamics model construction module and nonlinear dynamics model construction module can be embedded in a computer processing system. The computer calls the above-mentioned modules to complete the task of dynamic modeling of the planetary gear system based on the above-mentioned nonlinear dynamics modeling method of the gear transmission system considering thermal deformation; the above-mentioned receiving module, tooth surface flash temperature model construction module, tooth profile thermal deformation model construction module, bearing dynamics model construction module and nonlinear dynamics model construction module can perform operations according to the specific steps given in the above-mentioned nonlinear dynamics modeling method of the gear transmission system considering thermal deformation.

[0227] It should be noted that it should be understood that the division of the various modules of the above system is only a division of logical functions. In actual implementation, they can be fully or partially integrated into a physical entity, or they can be physically separated. Moreover, these modules can all be implemented in the form of software called by processing elements; or they can all be implemented in the form of hardware; or some modules can be implemented in the form of software called by processing elements, and some modules can be implemented in the form of hardware. For example, the receiving module can be a separately established processing element, or it can be integrated into a chip of the above-mentioned device. In addition, it can also be stored in the memory of the above-mentioned device in the form of program code, and called by a processing element of the above-mentioned device to perform the functions of the above-mentioned signal processing module. The implementation of other modules is similar. In addition, these modules can all or partly be integrated together, or they can be implemented independently. The processing element described here can be an integrated circuit with signal processing capabilities. In the implementation process, each step of the above method or each of the above modules can be completed by the hardware integrated logic circuit in the processor element or by instructions in the form of software.

[0228] For example, the above modules may be one or more integrated circuits configured to implement the above methods, such as one or more application specific integrated circuits (ASICs), or one or more microprocessors (digital signal processors).

[0229] Processor, referred to as DSP), or one or more field programmable gate arrays (Field

[0230] Programmable Gate Array (FPGA), etc. For another example, when one of the above modules is implemented by scheduling program code via a processing element, the processing element can be a general-purpose processor, such as a central processing unit (CPU) or other processor capable of calling program code. For another example, these modules can be integrated together and implemented as a system-on-a-chip (SOC).

[0231] Example 3:

[0232] The present invention provides a terminal device, comprising a memory, a processor, and a computer program stored in the memory and capable of running on the processor. When the processor loads and executes the computer program, the above-mentioned nonlinear dynamic modeling method of the gear transmission system considering thermal deformation is adopted.

[0233] It should be noted that the terminal device can be a computer device such as a desktop computer, a laptop computer or a cloud server, and the terminal device includes but is not limited to a processor and a memory. For example, the terminal device can also include input and output devices, network access devices and buses, etc.

[0234] Furthermore, the processor may adopt a central processing unit (CPU). Of course, depending on the actual usage, other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. may also be adopted. The general-purpose processor may adopt a microprocessor or any conventional processor, etc., and this application does not impose any restrictions on this.

[0235] Example 4:

[0236] The present invention provides a storage medium containing computer executable instructions, which are used to execute the above-mentioned nonlinear dynamic modeling method of a gear transmission system considering thermal deformation when executed by a computer processor.

[0237] Among them, the computer program can be stored in a computer-readable medium, the computer program includes computer program code, the computer program code can be in the form of source code, object code, executable file or certain middleware, etc. The computer-readable medium includes any entity or device that can carry computer program code, recording medium, USB flash drive, mobile hard disk, magnetic disk, optical disk, computer memory, read-only memory (ROM), random access memory (RAM), electrical carrier signal, telecommunication signal and software distribution medium, etc. It should be noted that computer-readable medium includes but is not limited to the above-mentioned components.

[0238] Embodiment 5:

[0239] The present invention provides a computer program product, which includes a computer program. When the computer program is executed by a processor, it is used to load and execute the above-mentioned nonlinear dynamic modeling method of a gear transmission system considering thermal deformation.

[0240] It should be noted that, in this document, relational terms such as first and second, etc., are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprises," "comprising," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that includes a list of elements includes not only those elements but also other elements not explicitly listed, or elements inherent to such process, method, article, or apparatus.

[0241] For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to the specific circumstances. When an element is referred to as being "assembled on", "installed on", "fixed on" or "set on" another element, it can be directly on the other element or there can be a central element. When an element is considered to be "connected" to another element, it can be directly connected to the other element or there can be a central element at the same time. The terms "vertical", "horizontal", "up", "down", "left", "right" and similar expressions used herein are for illustrative purposes only and are not intended to be the only embodiment.

[0242] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to these embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the appended claims and their equivalents.

[0243] Throughout this specification, references to terms such as "one embodiment," "example," or "specific example" indicate that a specific feature, structure, material, or characteristic described in conjunction with that embodiment or example is included in at least one embodiment or example of the present disclosure. In this specification, schematic representations of these terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

Claims

1. A nonlinear dynamic modeling method for a gear transmission system considering thermal deformation, characterized in that: The following steps are involved: receiving component parameters and operating parameters of the planetary gear system; Based on the component parameters and operating parameters of the planetary gear system, a tooth surface flash temperature model is constructed to calculate the tooth surface friction coefficient, and the tooth surface friction force and friction torque are calculated based on the tooth surface friction coefficient; Construct a tooth profile thermal deformation model and calculate the internal excitation of the planetary gear system considering the thermal deformation of the tooth profile, including the meshing error calculation, time-varying meshing stiffness calculation, and tooth side clearance calculation; Construct a bearing dynamics model and calculate the bearing clearance displacement function and bearing support force considering temperature rise; The internal excitation considering the temperature rise effect is introduced into the planetary gear system, and combined with the bearing dynamics model, a nonlinear dynamic model of the helical planetary gear train-bearing system is established.

2. The nonlinear dynamic modeling method of a gear transmission system considering thermal deformation according to claim 1, characterized in that: The planetary gear system includes a sun gear, planet gears, an inner ring gear, a planet carrier and bearings; The component parameters of the planetary gear system include mass, number of planetary gears, number of teeth on the inner ring gear, module of the inner ring gear, tooth width, pitch circle radius, helix angle, pressure angle, base circle radius of each gear, and moment of inertia of each gear; The operating parameters of the planetary gear system include input speed and torque, initial temperature, and operating temperature.

3. The nonlinear dynamic modeling method of a gear transmission system considering thermal deformation according to claim 2, characterized in that: Based on the component parameters and operating parameters of the planetary gear system, a tooth surface flash temperature model is constructed to calculate the tooth surface friction coefficient. The tooth surface friction force and friction torque are calculated based on the tooth surface friction coefficient, as follows: The calculation model of tooth surface instantaneous flash temperature is obtained based on Blok flash temperature theory: Where u is the friction coefficient, f m is the temperature rise coefficient, f e is the normal load per unit tooth width on the tooth surface, v1 and v2 are the tangential velocities of the meshing line of the driving wheel and the driven wheel respectively, g1 and g2 are the heat transfer coefficients of the driving wheel and the driven wheel respectively, ρ1 and ρ2 are the material densities of the two gears respectively, c1 and c2 are the specific heat capacities of the two gear materials respectively, and B is half of the contact band width; The tangential velocity calculation expression of the meshing line between the driving wheel and the driven wheel is as follows: Where, ω i Represents the angular velocity of the driving wheel and the driven wheel, r i They represent the pitch circle radius of the driving wheel and the driven wheel respectively, α represents the initial pressure angle of the gear, r ki is the distance from the meshing point to the center of the driving wheel and the driven wheel, where 1 represents the driving wheel and 2 represents the driven wheel; The tooth surface friction coefficient refers to the ratio of friction force to normal pressure during gear meshing. Due to differences in lubrication conditions, the friction coefficient will change dynamically. In actual high-temperature and high-speed working conditions, the planetary gear system is in a mixed lubrication state. Mixed lubrication states include dry friction, boundary lubrication, mixed lubrication, and elastohydrodynamic lubrication. The tooth surface friction coefficient in the mixed lubrication state is: in, v s (t)=|v1(t)-v2(t)| Where S av is the average value of tooth surface roughness, S1 and S2 are the surface roughness of the driving wheel and the driven wheel respectively, P ei (t) is the normal load on the gear tooth width, T is the total load on the tooth surface, b i are the tooth widths of the driving wheel and the driven wheel, r bi are the base circle radii of the driving wheel and the driven wheel respectively, are the entrainment speeds of the driving wheel and the driven wheel, v s (t) is the relative sliding speed between the driving wheel and the driven wheel, and η0 is the dynamic viscosity coefficient of the lubricating oil.

4. The nonlinear dynamic modeling method of a gear transmission system considering thermal deformation according to claim 3, characterized in that: A tooth profile thermal deformation model is constructed to calculate the internal excitation of the planetary gear system considering the thermal deformation of the tooth profile, including meshing error calculation, time-varying meshing stiffness calculation, and tooth backlash calculation. Specifically, the following are included: (41) The tooth profile thermal deformation calculation model is a gear thermal deformation prediction method established based on the principles of thermodynamics and material mechanics, combined with numerical simulation technology. By analyzing the thermal expansion characteristics and temperature field distribution law of gear materials at high temperatures, it reflects the nonlinear relationship between the material linear expansion coefficient and the temperature gradient. The calculation formula for tooth profile thermal deformation is as follows: in, Where Δf is the thermal deformation of the tooth profile, Δl is the radius r k , the angle is An arc, α is the initial pressure angle, is the angle between the meshing point and the gear center before and after deformation, r k is the distance between the meshing point and the gear center before deformation, λ is the thermal expansion coefficient, is the instantaneous contact temperature at the meshing point, T0 is the initial temperature, s is the tooth thickness at the pitch circle, r is the pitch circle radius, α k is the pressure angle of the meshing point after deformation, r k' is the distance between the meshing point and the gear center after deformation, r b is the base circle radius, u b is the distance between the pitch circle and the base circle, r x is the radial distance, is the radial distance r x The instantaneous contact temperature, r a is the radial distance of the tooth tip, r s is the radial distance at the pitch circle; (42) Meshing error excitation calculation considering thermal deformation: When the planetary gear system operates under high temperature conditions, the significant increase in tooth surface temperature will trigger the thermal deformation effect of the material, which in turn leads to the nonlinear offset of the involute geometric characteristics of the tooth profile. The tooth profile error caused by the temperature increase of the tooth surface is expressed as: Where s gi (i=1,2) represents the tooth thickness of the two gears. The external meshing pair driving gear is the sun gear, and the internal meshing pair driving gear is the planet gear. bi Indicates the base circle radius of the two gears, r ci represents the curvature radius of the two gears, u bi is the thermal expansion deformation of the two gears, α ki is the pressure angle of the contact point after thermal expansion of the gear; The comprehensive tooth profile deviation considering the tooth surface temperature rise and manufacturing error can be expressed as: Where, represents the initial tooth profile error of the two gears due to manufacturing error, (G T+ΔT ) 1,2 Indicates the deviation of the tooth profiles of the two gears caused by temperature rise; The obtained comprehensive tooth profile deviation is equivalent to the comprehensive equivalent error along the meshing line of the gear pair: The meshing error excitation of the planetary gear system is obtained as: Where, E s 、E pi are the eccentricity errors of the sun gear and the i-th planet gear, ω s 、ω pi 、ω c are the rotation frequencies of the sun gear, the i-th planet gear and the planet carrier, β s , β pi are the initial phases of the eccentricity errors between the sun gear and the i-th planet gear, φ pi is the position angle of the i-th planetary gear, β b is the helix angle; (43) Calculation of gear meshing deformation coordination conditions considering thermal deformation: Based on the elastic deformation characteristics of the internal and external meshing in the planetary gear system, a deformation coordination equation considering material elastic deformation, meshing error and vibration displacement is established to solve the relative displacement of the internal and external gear meshing pairs on the meshing line. The equivalent displacement of the sun gear and planetary gear meshing pairs is expressed by δ spi express: d spi =(-x s sinψ spi +y s cosψ spi +u s +x pi sinψ spi -y pi cosψ spi +u pi ) cosβ b +(with s -With pi )sinβ b -e spi Where x s 、y s 、z s and u s are the displacement of the sun gear along the coordinate system and the torsion angle around the z axis, x pi 、y pi 、z pi and u pi are the displacement of the planetary gear along the coordinate system and the torsion angle around the z axis, ψ spi is the angle between the meshing plane of the external meshing pair and the y-axis, β b is the helix angle, e spi is the meshing error of the external gear pair; The equivalent displacement of the meshing pair of planetary gear and internal gear ring is expressed as δ rpi express: d rpi =(x r sinψ rpi -y r cosψ rpi +u r -x pi sinψ rpi +y pi cosψ rpi -u pi )cosβ b +(z pi -z r )sinβ b -e rpi Where x r 、y r 、z r and u r are the displacement of the inner gear ring along the coordinate system and the torsion angle around the z axis, ψ rpi is the angle between the meshing plane of the internal meshing pair and the y-axis, e rpi Meshing error of internal gear pair; The equivalent displacement projection of the planet gear along the planet carrier coordinate system is: Where, δ cpix , δ cpiy , δ cpiz and δ cpiu are the projections of the equivalent displacement of the planetary gear along the planetary carrier coordinate system, x c 、y c 、z c and u c are the displacement of the planet carrier along the coordinate system and the torsion angle around the z axis, φ pi is the planet gear position angle; (44) Calculation of time-varying mesh stiffness considering gear thermal deformation: The potential energy method is used to calculate the time-varying mesh stiffness of gears; (44.1) During the meshing process, a pair of gears will experience single-tooth meshing and double-tooth meshing respectively. When the gear is in the single-tooth meshing region, the calculation formula for the time-varying mesh stiffness is: Where k eh (t) is the Hertzian contact stiffness, k ebi (t)(i=1,2) represents the bending stiffness of the driving wheel and the driven wheel respectively, k esi (t)(i=1,2) represents the shear stiffness of the driving wheel and the driven wheel respectively, k eai (t)(i=1,2) represents the radial compression stiffness of the driving wheel and the driven wheel respectively, k efi (t) (i=1, 2) represents the matrix stiffness of the driving wheel and the driven wheel respectively; (44.2) When the gear is in the double-tooth meshing region, the calculation formula for the time-varying mesh stiffness is: Where k eh,j (t)(j=1, 2) respectively represent the Hertzian contact stiffness of the two meshing pairs when the driving wheel and the driven wheel are in the double-tooth meshing area; As the gear temperature rises during operation, the instantaneous contact temperature of the tooth surface changes, causing the tooth surface profile to deform and the meshing stiffness to change. According to Hertz contact theory, the stiffness change of the driving and driven wheels caused by the instantaneous contact temperature change of the gear surface is: Where k Ti (i=1, 2) represents the change of the tooth surface stiffness of the driving wheel and the driven wheel respectively, b i (i=1, 2) represents the tooth width of the driving wheel and the driven wheel respectively, Δf is the thermal deformation of the tooth profile, F n is the tooth surface pressure, and the stiffness changes of the two tooth surfaces in series can be obtained by the equivalent stiffness k caused by temperature change. T (t); Assuming that the thermal deformation of the internal and external meshing of the planetary gear train is consistent, the calculation formula for the time-varying meshing stiffness considering the thermal deformation of the gear can be obtained as follows: k spi (t)=k rpi (t)=k e (t)+k T (t) Where k spi (t) is the time-varying meshing stiffness of the planetary gear system, k rpi (t) is the time-varying meshing stiffness of the planetary gear train, k e (t) is the time-varying mesh stiffness without considering the temperature rise, k T (t) is the equivalent stiffness caused by temperature change; (45) Calculation of tooth side clearance considering thermal deformation: tooth side clearance refers to the minimum clearance between the two gear tooth sides in the direction perpendicular to the axis when the gear pair is in the non-meshing state. Since the gear teeth will produce thermal deformation during the transmission process, the tooth side clearance of the gear will decrease. Therefore, after considering the influence of thermal deformation, the tooth side clearance value calculation formula is: Where b' is half of the tooth backlash value after considering thermal deformation, b is half of the initial value of the tooth backlash, Δf1 and Δf2 are the thermal deformation of the tooth profile of the driving wheel and the driven wheel respectively; The calculation formula for tooth side clearance taking thermal deformation into account is: Where x is the displacement along the gear meshing line.

5. The nonlinear dynamic modeling method of a gear transmission system considering thermal deformation according to claim 4, characterized in that: A bearing dynamics model was constructed to calculate the bearing clearance displacement function and bearing support force considering the temperature rise, as follows: The nonlinear displacement function of the bearing clearance considering the thermal deformation of the rolling elements is: in, b bi =b b0 -Δf b Where x bi with y bi The clearance displacement of the two bearings along the two directions of the coordinate system, b bi is half of the clearance value after thermal deformation of the two bearings, b b0 is half of the initial bearing clearance value, Δf b is the thermal deformation of the rolling element; The bearing support force is the resultant of the forces acting on each roller. The resultant bearing support forces in the x and y directions are obtained by projecting the normal force at the contact point of each rolling element into its radial direction and then solving and summing the radial forces of all rolling elements in the x and y directions. The forces acting on each rolling element are calculated using Hertz contact theory: Where K b is the rolling element stiffness, Δ bk is the deformation of the kth rolling element, H(Δ bk ) is the Hertzian contact function; Decompose each rolling element load along the x and y directions, and sum them to obtain the bearing support force along the two directions: Where, ψ' b is the contact angle, θ bk is the angular displacement of the kth rolling element.

6. The nonlinear dynamic modeling method of a gear transmission system considering thermal deformation according to claim 5, characterized in that: In the nonlinear dynamic model of the helical planetary gear train-bearing system, both the internal gear and the external gear are involute helical gears, each component is simplified as a rigid body, and the installation errors between the components are ignored; the contact mode of the joint surface is simplified to a spring and damper connection; The planetary gears are evenly distributed along the circumference, and the average meshing stiffness, support stiffness, mass, moment of inertia and structural dimensions of the planetary gears are all equal.

7. A nonlinear dynamic modeling system for a gear transmission system taking thermal deformation into consideration, for implementing a nonlinear dynamic modeling method for a gear transmission system taking thermal deformation into consideration according to any one of claims 1 to 6, characterized in that: include: A receiving module, configured to receive component parameters and operating parameters of the planetary gear system; The tooth surface flash temperature model construction module is used to construct the tooth surface flash temperature model and calculate the tooth surface friction coefficient based on the component parameters and operating parameters of the planetary gear system, and calculate the tooth surface friction force and friction torque based on the tooth surface friction coefficient; The tooth profile thermal deformation model construction module is used to construct the tooth profile thermal deformation model and calculate the internal excitation of the planetary gear system considering the thermal deformation of the tooth profile, including the meshing error calculation, time-varying meshing stiffness calculation, and tooth side clearance calculation; The bearing dynamics model construction module is used to construct the bearing dynamics model and calculate the bearing clearance displacement function and bearing support force considering the temperature rise; The nonlinear dynamic model building module is used to introduce internal excitation considering the temperature rise effect into the planetary gear system, and combine it with the bearing dynamic model to establish a nonlinear dynamic model of the helical planetary gear train-bearing system.

8. A terminal device comprising a memory, a processor, and a computer program stored in the memory and capable of running on the processor, characterized in that: When the processor loads and executes the computer program, the nonlinear dynamic modeling method of the gear transmission system considering thermal deformation according to any one of claims 1 to 6 is adopted.

9. A storage medium containing computer-executable instructions, characterized in that: When executed by a computer processor, the computer executable instructions are used to perform the nonlinear dynamic modeling method of a gear transmission system considering thermal deformation according to any one of claims 1 to 6.

10. A computer program product, characterized in that The computer program product includes a computer program, and when the computer program is executed by a processor, the computer program is used to load and execute the nonlinear dynamic modeling method of a gear transmission system considering thermal deformation according to any one of claims 1 to 6.

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